arX

iv:1

901.

1131

2v1

[gr

-qc]

31

Jan

2019 Spin connection formulations of real

Lorentzian General Relativity

Ermis Mitsou

Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, Universityof Zurich, CH–8057 Zurich, Switzerland

E-mail: ermitsou@physik.uzh.ch

Abstract. We derive the pure spin connection and constraint-free BF formulations of real four-dimensional Lorentzian vacuum General Relativity. In contrast to the existing complex formulations,an important advantage is that they do not require the reality constraints that complicate quantiza-tion. We also consider the corresponding modified gravity theories and point out that, contrary totheir self-dual analogues, they are not viable because they necessarily contain ghosts. In particular,this constrains the set of potentially viable unified theories one can build by extending the gaugegroup to the ones with the action structure of General Relativity. We find, however, that the result-ing theories do not admit classical solutions. This issue can be solved by introducing extra dynamicalfields which, incidentally, could also provide a way to include a matter sector.

Contents

1 Introduction & summary 1

2 Notation and useful identities 3

3 Plebanski actions 5

4 Derivation of the pure connection action 6

5 Derivation of the constraint-free BF action 9

6 The SL(2,C) and SD formulations 11

7 The linearized pure connection action over (A)dS 12

8 No go for real modified actions 15

9 Group extensions and unification 17

1 Introduction & summary

Alternative formulations of classical General Relativity (GR) provide us with important insights aboutthe theory, new perspectives for potential modifications or extensions and also genuinely new startingpoints for approaching the quantum theory. One line of research in this direction are the so-called“pure connection” formulations, where the usual gravitational field that is a metric or a vierbein,is replaced by a connection associated with some group, in close analogy with the mathematicaldescription we have for the rest of the known forces of nature. In fact, this suggestive resemblance is animportant motivation for this approach, because enlarging the gauge group leads to extra connectioncomponents and therefore constitutes an elegant potential path towards unification (see the review[1] and references therein). If one considers the metric formalism, then the only available group isthe diffeomorphism group, so extending it necessarily introduces extra dimensions. In the vierbeinformalism, however, we have the internal action of the Lorentz group, so the latter can be extendedwithout altering the dimensionality of space-time.

Nevertheless, it is instructive to quickly expose the metric case, whose development dates backto Eddington [2] and Schrodinger [3]. The starting point is the Hilbert-Palatini action for a metricgµν and an independent symmetric affine connection Γρ

µν ≡ Γρνµ

S =1

16πG

∫d4x

√−g (gµνRµν − 2Λ) , (1.1)

where

Rµν := ∂ρΓρµν − ∂(µΓ

ρν)ρ + Γρ

ρσΓσµν − Γρ

σµΓσρν . (1.2)

Integrating out Γρµν , i.e. replacing it with the solution to its equation of motion, yields the Einstein-

Hilbert action. On the other hand, integrating out gµν leads to the pure connection action

S → 1

8πGΛ

∫d4x

√− detRµν . (1.3)

Note that this manipulation requires Λ 6= 0, which fits within the present concordance viewpoint oncosmology and leads in particular to a tiny dimensionless “coupling constant”

√8πGΛ ∼ 10−60 in

~ = c = 1 units.

– 1 –

In the vierbein formalism, the situation is more involved. If one starts with the analogous Hilbert-Palatini formulation, with the independent connection being now associated with the Lorentz group(the “spin” connection), then it is easy to see that the vierbein cannot be isolated in its equationof motion in an algebraically simple manner. A more fruitful starting point is rather the Plebanskiaction [4], which has led to a formulation involving the spin connection and an auxiliary scalar densityby Capovilla, Jacobson and Dell [5–7] (see also [8, 9]) and more recently to a formulation involving thespin connection alone for the Λ 6= 0 case by Krasnov [10, 11] (see also [12–15] for the correspondingquantum field theory). One common feature of these formulations is that they depend on the self-dual component of the spin connection alone. They can therefore be seen as gauge theories of thecomplexified group SO(3,C), commonly referred to as “complex GR”, and thus require “reality”constraints on the curvature field in order to select the sector corresponding to real Lorentzian GR.These constraints depend on the action and field content and are especially difficult to handle in thequantum theory. We will refer to these formulations as the “self-dual” ones (SD).

In the case of real GR, where the full spin connection is considered, the algebraic structure ismore complicated and this fact has prevented the derivation of the pure connection action so far.In this paper we will show how to overcome this difficulty and will therefore obtain the pure spinconnection formulation of vacuum GR with Λ 6= 0. As in the SD case [10, 11], the action involves amatrix square root

SGR[F ] :=1

16πGΛ

∫Tr[√zXz

]2, (1.4)

where

X abcd := F ab ∧ Fcd , zabcd :=

(γ + 1)(δac δ

bd − δadδ

bc

)+ (γ − 1) εabcd

2γ, (1.5)

are seen as a 6 × 6 matrices in antisymmetric pairs of Lorentz indices [ab], F ab are the curvature2-forms, γ is the Immirzi parameter, εabcd is the Levi-Civita symbol and the indices are displacedwith the Minkowski metric ηab. Note that in the Λ = 0 case (1.4) can be expressed non-singularlywith the help of an extra Lagrange multiplier density, but the corresponding dynamics are not welldefined, contrary to the SD formulation [6].

Another useful approach to gauge theories is through the so-called “BF” or “covariant Hamilto-nian” formulation, where one integrates in an auxiliary set of 2-forms Bab to make the equations ofmotion first-order in derivatives. The best known BF formulation of GR is the Plebanski action, inboth complex [4] and real [16] forms, in which case one requires an extra set of Lagrange multipli-ers in order to impose some conditions on Bab known as the “simplicity” or “metricity” constraints.Recently, however, it was noted by Herfray and Krasnov [17, 18] that there exists a constraint-freeBF formulation of GR, at least in the real case, since the complex one still requires the reality con-straints. This description is therefore closer to the structure of standard gauge theory, which is againan important feature if one is interested in unified theories. Due to the same kind of algebraic com-plications that prevented the derivation of the real pure connection action, the real version of this BFformulation has not yet been given either. Here we derive this action too, with the result being

SGR[B,F ] :=1

2

∫ Bab ∧ F ab +Tr (wYw) +

[Tr

√zw2Yw2z

]2

8πGΛ− Tr[(zw)2

]

, (1.6)

where nowYab

cd := Bab ∧Bcd , wabcd := w

(δac δ

bd − δadδ

bc

)+ w εabcd , (1.7)

and w, w are two additional free parameters. In contrast to the pure connection formulation (1.4),here we do have access to the Λ = 0 case.

By expressing (1.4) and (1.6) in their SL(2,C) form, one can recover the SD formulations [10,11, 17, 18] either by simply setting γ = −i, or by imposing the appropriate reality constraints. Indoing so, we will clarify how an implicit sign ambiguity in the scalar reality constraint, for the pureconnection case, is related to the two sectors of the underlying Plebanski theory. Another interestingcomputation in this paper is the linearization of the pure connection theory around (anti-)de-Sitter

– 2 –

space-time ((A)dS). We perform it for generic γ values and find agreement with the earlier perturbativeconstructions for the non-chiral case |γ| → ∞ [19, 20]. A special aspect of the quadratic action isthat it only depends on the fully traceless part of the curvature tensor and we show that this is apeculiarity of the GR action.

Finally, we will also consider the known generalizations of (1.4) and (1.6) where the matrixfunction f(M) = [Tr

√M ]2 is replaced by an arbitrary homogeneous function of degree one [21–29].

We will argue that, contrary to the SD case, these modified gravity theories have an unavoidablepathology in that one of their two gravitons is necessarily a ghost. This fact, which does not seemto be noted in the literature to our knowledge, makes it all the more important to know the precisematrix functions (1.4) and (1.6) that correspond to GR. This is also relevant for the correspondingunified theories, i.e. those obtained by extending the gauge group SO(1, 3) → G, because then onlythe GR Lagrangian function has a chance of yielding viable results. We will show, however, that thecorresponding equations of motion admit no solution as soon as the dimension of G exceeds six. Thissituation could be avoided by introducing extra dynamical fields which, incidentally, could also allowone to describe the matter sector.

The paper is organized as follows. In section 2 we define a compact matrix notation along withan algebraic toolkit that will make our computations much more straightforward and transparent. Insection 3 we point out the equivalence between a nine-parameter family of real Plebanski formulations,which is useful because some of them can be more suited than others, depending on the problem athand. We then derive the pure-connection action in section 4 and the constraint-free BF action insection 5. In section 6 we show the relation with the SD formulations and in section 7 we derive thelinearized pure connection theory. Finally, in section 8 we discuss the ghost problem of the modifiedgravity theories and in section 9 we consider some aspects of the group extension of the GR action.

2 Notation and useful identities

We will work with tensors in the Lorentz algebra that are therefore indexed by antisymmetric pairs ofLorentz indices [ab] forming a six-dimensional index. More precisely, there will be vectors V [ab] and2-tensors M [ab][cd] and the metric displacing these pairs of indices is the Killing form of the algebra

κ[ab][cd] := ηacηbd − ηadηbc . (2.1)

We can therefore introduce a compact notation in which V ab → |V 〉 is a vector and Mabcd →M is a

matrix. The product and trace definitions must then include 1/2 factors for each contraction

(M |V 〉)ab := 1

2Mab

cdVcd , (M2)abcd :=

1

2Mab

efMefcd 〈M〉 := 1

2Mab

ab , (2.2)

to avoid counting every independent component twice. The Lorentz-compatible transposition opera-tion is

|V 〉κ := |V 〉Tκ ≡ 〈V | , 〈V |κ ≡ |V 〉 , Mκ := κ−1MTκ , (2.3)

which therefore yields covectors Vab when acting on vectors V ab, and vice-versa, and exchanges thepairs of indices for tensors (Mκ)abcd ≡Mcdab. We will refer to it simply as “transposition” and notethat it has all the algebraic properties of the usual matrix transposition. In particular, we can definethe corresponding “symmetric” and “antisymmetric” matrices as the ones satisfying

Sκ ≡ S , Aκ ≡ −A , (2.4)

respectively, and any matrix M can be decomposed as

M ≡ S +A , S :=1

2(M +Mκ) , A :=

1

2(M −Mκ) . (2.5)

We can also define two Lorentz-invariant inner products and an outer one

〈V |W 〉 ≡ 〈W |V 〉 ≡ 1

2VabW

ab , 〈MκN〉 ≡ 1

4MabcdN

abcd , (|V 〉〈W |)abcd ≡ V abWcd , (2.6)

– 3 –

with the inner ones being invariant under transposition, while (|V 〉〈W |)κ ≡ |W 〉〈V |. There are twoLorentz-invariant matrices, the identity and the dual operator (or “complex structure”)

1abcd := κabcd ≡ δac δ

bd − δadδ

bc , ⋆abcd := εabcd , (2.7)

respectively, which are symmetric and satisfy

1M ≡M1 ≡M , ⋆2 ≡ −1 , 〈1〉 ≡ 6 , 〈⋆〉 ≡ 0 . (2.8)

With the identity operator one can define the inverse M−1 of a matrix M

M−1M ≡ 1 ⇔ 1

2(M−1)abefM

efcd ≡ δac δ

bd − δadδ

bc . (2.9)

Observe that the real matrices of the form z := α1 + β ⋆ have the algebraic behavior of complexnumbers among themselves, e.g.

zz′ ≡ z′z ≡ (αα′ − ββ′)1+ (αβ′ + α′β) ⋆ , z−1 ≡ α1− β ⋆

α2 + β2, (2.10)

with the only important difference being that they do not commute with other matrices in general.We will refer to such matrices as the “invariant” ones. Every symmetric matrix S can be decomposedinto an invariant part and a covariant one

S ≡ 1

6

(S 1− S ⋆

)+ S , S := 〈S〉 , S := 〈⋆S〉 , 〈S〉 ≡ 〈⋆S〉 ≡ 0 , (2.11)

or, equivalently, into scalar and pseudo-scalar traces and a traceless part. We next note that thepresence of a complex structure ⋆ provides a “conjugation” operation on matrices

M⋆ := − ⋆M ⋆ , (2.12)

which is also a symmetry of the matrix inner product (2.6) and of the invariant matrices 1 and ⋆.Moreover, it commutes with transposition Mκ⋆ ≡ M⋆κ, it is an involution M⋆⋆ ≡ M and also asimilarity transformationM⋆ ≡ ⋆−1M ⋆, meaning that it commutes with any analytic matrix functionf(M)⋆ ≡ f(M⋆). We can then decompose any matrix into its “even” and “odd” components

M ≡M+ +M− , M⋆± ≡ ±M± M± :=

1

2(M ±M⋆) , (2.13)

which can be alternatively stated as

[ ⋆,M+] ≡ 0 , {⋆,M−} ≡ 0 , (2.14)

respectively. We also note that, as in the case of antisymmetric matrices A, the odd ones M− haveno invariant component 〈M−〉 ≡ 〈⋆M−〉 ≡ 0. Expressing this conjugation in terms of Lorentz indiceswe find

(M⋆)abcd ≡Mcdab − ηcaMdb + ηcbMda + ηdaMcb − ηdbMca , (2.15)

where

Mab :=M cacb −

1

4ηabM

abab , (2.16)

is identically traceless. Therefore, for a symmetric matrix S

(S+)abcd ≡ Sabcd +1

6εabcd S − ηa[cSd]b − ηb[cSd]a (2.17)

≡ 1

3

[Sabcd +

1

2(Sacbd − Sadbc − Sbcad + Sbdac) + Scdab

]− ηa[cSd]b − ηb[cSd]a ,

– 4 –

is the fully traceless component (S+)cacb ≡ 0 with the symmetries of the Weyl tensor, while

(S−)abcd ≡ (S−)abcd ≡ ηa[cSd]b − ηb[cSd]a , (2.18)

is only partially traceless (S−)cacb ≡ Sab, therefore capturing the 2-tensor part of Sabcd. These roles

are switched for antisymmetric matrices, i.e. the fully traceless part is the odd component A−

(A−)abcd ≡ Aabcd + ηa[cAd]b + ηb[cAd]a ⇒ (A−)cacb ≡ 0 , (2.19)

while the even component A+ is the partially traceless one

(A+)abcd ≡ −ηa[cAd]b + ηb[cAd]a ⇒ (A+)cacb ≡ Aab . (2.20)

Thus, a generic matrix M can be uniquely decomposed as

M ≡ 1

6

(S 1− S ⋆

)+ S+ + S− +A+ +A− , (2.21)

with every term having a definite behavior under both transposition and conjugation and beingnormal to all others with respect to the matrix inner product (2.6). Moreover, notice that each ofthese components corresponds to an irreducible representation of the Lorentz group. In contrast tothe usual definitions involving index symmetries, we see that in four dimensions the presence of ⋆allows one to define these components in a more algebraic way, namely, by distinguishing among allthe possible behaviors under matrix trace, transposition and conjugation.

Finally, the (co-)vectors and matrices described above will always be differential forms of evendegree, and it is understood that their products will always be wedge products, so their matrix/vectorcomponents commute. Also, when manipulating the Lagrangian, we will sometimes consider rationalpowers of 4-forms, as in (1.4) for instance. These should therefore be interpreted as powers of thecorresponding weight one densities, with the d4x factors always appearing with the correct power atthe end to make the result coordinate-independent.

3 Plebanski actions

Here by “Plebanski Lagrangian” we will mean a member of the following family

L = 〈B|z1|F 〉 −1

2〈B| (z2 + z3ψz3) |B〉+ φ [〈z4ψ〉 − α] , (3.1)

where

F ab := dAab +Aac ∧ Acb , (3.2)

are the curvature 2-forms of a real spin connection 1-formAab, while ψabcd, Bab and φ are real auxiliary0, 2, and 4-forms, respectively, and ψ is symmetric. Moreover, the zk are real invariant matrices withconstant coefficients. Out of the nine parameters present in (3.1), only two are relevant, because sevenof them can be eliminated by linear/affine redefinitions of the auxiliary fields

φ→ α′φ , |B〉 → z′1|B〉 , ψ → z′2 + z′3ψz

′3 , (3.3)

and these relate all the possible formulations found in the literature [16, 30–33]. The two relevantparameters are ultimately λ := 8πGΛ and γ that were mentioned in the introduction. Observe,however, that the equivalence of all these Lagrangians is a feature of the Lorentzian case only, becausein the Euclidean case, where ⋆2 ≡ 1, not all real z 6= 0 are invertible. Note also that the constraintimposed by φ on ψ is sometimes considered implicitly, i.e. without including the corresponding termin the Lagrangian.

– 5 –

4 Derivation of the pure connection action

We begin with the vierbein Hilbert-Palatini-Holst Lagrangian 4-form of vacuum GR with mostly-pluses signature

L =1

8πG

[〈E|

(1

γ+ ⋆

)|F 〉 − Λ e

], (4.1)

where

Eab := ea ∧ eb , e :=1

6〈E| ⋆ |E〉 ≡ 1

4!εabcd e

a ∧ eb ∧ ec ∧ ed , (4.2)

ea are the vierbein 1-forms and Aab are the (independent) spin connection 1-forms. Keep in mindthat it is also possible to include the topological terms 〈F |w|F 〉, where w is some constant invariantmatrix, which will only affect the quantum theory, but we neglect these here.

The case |γ| → ∞ corresponds to the standard Hilbert-Palatini action, which is parity-symmetric(“non-chiral”), while a finite γ leads to the inclusion of the parity-violating (“chiral”) Holst term[34]. As in the metric case, by integrating out Aab one recovers the Einstein-Hilbert action for themetric g := ηab e

a ⊗ eb. In this process, the relation between the vierbein and the spin connectionis independent of γ and the Holst term ∼ 〈E|F 〉 vanishes identically, so this extension does notinfluence the classical dynamics. In fact, if the theory is quantized in a Lorentz-covariant manner,then the physical observables seem to be independent of γ in the quantum theory too [35, 36]. Thiswas definitely settled in [37], where it was shown that with the appropriate parametrization of phasespace the γ dependence drops already at the level of the canonical action.

The theory (4.1) can be obtained as a sector of the Plebanski theory (3.1) and here a convenientchoice will be

L = 〈B|z|F 〉 − 1

2〈B|ψ|B〉+ φ [〈ψ〉 − λ] , λ := 8πGΛ , (4.3)

where

z(γ) :=(γ + 1)1+ (γ − 1) ⋆

2γ. (4.4)

The relation to (4.1) is obtained through the equation of motion of ψ, i.e. the “simplicity” constraints

|B〉〈B| = 2φ1 , (4.5)

which are solved by either

|B〉 = ± 1+ ⋆

8πG|E〉 , φ =

e

(8πG)2, (4.6)

or

|B〉 = ± 1− ⋆

8πG|E〉 , φ =

−e(8πG)2

, (4.7)

for some vierbein ea, where we have used

|E〉〈E| ≡ − ⋆ e . (4.8)

The solution (4.6) with the plus sign reproduces (4.1), while the one with the minus sign correspondsto ghost-like gravity, i.e. gravitons with negative kinetic energy. On the other hand, (4.7) leads tothe same dynamics (i.e. neglecting the Holst term) as (4.6) for finite γ, but with rescaled constants

G→ γG , Λ → −γΛ . (4.9)

In the |γ| → ∞ limit the kinetic term in the Lagrangian becomes pure Holst ∼ 〈E|F 〉, thus corre-sponding to a topological theory. For later reference, we will refer to (4.6) as the “right” sector and to(4.7) as the “wrong” sector, since our starting point is (4.1). The pure connection formulation beingbased on (4.3), it will necessarily contain all of these four solutions, i.e. the right and wrong sectorsalong with their respective sign ambiguities, and we will see how to distinguish among these later on.Note also that the |B〉 ∼ |E〉 relations (4.6) and (4.7) depend on the choice of Plebanski Lagrangian,but the relation (4.9) between the two sectors does not.

– 6 –

In complete analogy with the treatment of the SD case [5–11], we will now integrate out Bab,then ψabcd and, if Λ 6= 0, also φ. In that procedure, the non-trivial step which was resisting completionso far [18] is finding the solution of ψabcd. Plugging the solution to the equation of motion of Bab

|B〉 = ψ−1z|F 〉 , (4.10)

back inside the Lagrangian (4.3), we get

L→⟨1

2ψ−1X z + φψ

⟩− λφ , (4.11)

whereX z := zXz , X := |F 〉〈F | , (4.12)

are symmetric matrices of 4-forms. Next, given the inverse matrix variation rule

0 ≡ δ1 ≡ δ(ψψ−1

)≡ (δψ)ψ−1 +ψ δ(ψ−1) ⇒ δ(ψ−1) = −ψ−1(δψ)ψ−1 , (4.13)

and the cyclicity of the trace, the equation of motion of ψ is

2φψ2 = X z , (4.14)

whose solution is therefore simply

ψ = ±√

X z

2φ, (4.15)

with the choice of sign being irrelevant for the final action. Had we considered a different Plebanskiformulation (3.1), as was the case in earlier works [18], the equation of motion of ψ would have takenthe form

ψQψ = P , (4.16)

for two symmetric matrices Q,P , which is therefore not solved by simply taking a square root. Forinstance, we could have chosen (5.1), which is obtained from (4.3) by the redefinitions (5.2), andleads to Q = 2φz2 and P = X . Therefore, we see that by taking full advantage of the freedomin redefining the auxiliary fields of the Plebanski Lagrangian, we can considerably simplify somealgebraic manipulations. Nevertheless, it is instructive to note that one can actually proceed in thegeneral case too, i.e. that (4.16) also admits a closed-form symmetric solution

ψ = ±P 1/2(P 1/2QP 1/2

)−1/2

P 1/2 . (4.17)

In order to manipulate such expressions back inside the Lagrangian, however, it is useful to have inmind that (at least for invertible matrices) the cyclicity of the trace holds even in the presence of amatrix square root, because analytic matrix functions commute with similarity transformations

⟨√MM ′

⟩≡⟨MM−1

√MM ′

⟩≡⟨M−1

√MM ′M

⟩≡⟨√M−1MM ′M

⟩≡⟨√M ′M

⟩.

(4.18)Whichever the approach we choose, the resulting Lagrangian reads

L→ ±√2φ⟨√

X z

⟩− λφ , (4.19)

since it cannot depend on redefinitions of fields that have been integrated out. There are now twooptions: either λ = 0 or λ 6= 0. In the former case φ cannot be integrated out because it is not presentin its equation of motion, i.e. it acts as a Lagrange multiplier enforcing the constraint

⟨√X z

⟩= 0 . (4.20)

– 7 –

Moreover, the equation of motion of Aab involves the inverse matrix ∼ X−1/2z

, meaning that it requiresX z to be invertible. However, in the absence of a cosmological constant and sources, the maximallysymmetric solution is Minkowski space-time F ab = 0, meaning X z = 0, so one cannot describefluctuations around that solution. At the level of the action, this issue manifests itself as the fact thatthe square root is not differentiable at zero. In conclusion, contrary to the SD case1, the real pureconnection theory with λ = 0 is not well-defined in general.

Let us next consider the λ 6= 0 case, where now the φ field can be integrated out of (4.19).Plugging the solution to its equation of motion

√2φ = ± 1

λ

⟨√X z

⟩, (4.21)

back inside (4.19), we find the pure spin connection Lagrangian

L =1

2λ

⟨√X z

⟩2, (4.22)

hence the action (1.4). We can obtain an alternative form by using (4.18)

L ≡ 1

2λ

⟨√z2X

⟩2≡ 1

2λ

⟨√(1

γ1+

γ2 − 1

2γ2⋆

)X

⟩2

, (4.23)

which is simpler, although the matrix argument is no longer symmetric in general. In the non-chiralcase |γ| → ∞ we have

limγ→±∞

L =1

4λ

⟨√⋆X⟩2

, (4.24)

so the ⋆ combines with the Levi-Civita symbol hiding in the wedge product F ab ∧ F cb to yield aparity-symmetric Lagrangian indeed. To obtain the equations of motion of (4.22), we note that thevariation of an analytic matrix function inside a trace is simply δ〈f(M)〉 ≡ 〈f ′(M) δM〉, which isshown using the Taylor expansion and the cyclicity of the trace, and find

D[⟨√

X z

⟩X−1/2

zz]abcd

∧ F cd = 0 , (4.25)

where D is the exterior covariant derivative, we have used [D, ⋆] ≡ 0 and the Bianchi identity DF ab ≡ 0.Now the presence of the inverse of X z is no longer a problem, because the maximally symmetric solu-tion is (A)dS, for which X ∼ ⋆ is invertible, so we do have access to the dynamics of the correspondingfluctuations. More generally, we can describe any space-time for which X is invertible everywhere,since z is also invertible for real γ.

Let us finally come back to the issue of selecting the set of solutions corresponding to the originaltheory (4.1). As we will see in sections 6 and 7, it will sometimes be possible to distinguish betweenthe “right” and “wrong” sectors, but the action will be insensitive to the choice of sign within eachsector. This is because the two options are related by the sign flip |F 〉 → −|F 〉 at the level ofthe Hilbert-Palatini-Holst Lagrangian (4.1), and the pure connection one (4.22) only depends on thequadratic combination X := |F 〉〈F |, which is invariant. Nevertheless, this will not be a problem forpractical purposes. Indeed, given the relation (4.10) between |B〉 and |F 〉, we see that for |B〉 tochange sign |F 〉 must go through zero, meaning that X must become non-invertible, in which case thewhole construct breaks down anyway. Thus, restricting to solutions where X is invertible everywhereautomatically selects a definite sign within each sector.

1In the SD case with λ = 0 one can avoid this problem thanks to the lower dimensionality of the involved matrices[6]. The Lagrangian is formally (4.11), but the corresponding matrices are 3 × 3 complex symmetric. Thanks to this,by manipulating the characteristic equation of ψ−1 one can show that 〈ψ〉 ≡

[

〈ψ−1〉2 − 〈ψ−2〉]

/2 detψ, which makes(4.11) quadratic in ψ−1, after a redefinition of φ to absorb the determinant. Thus, integrating out ψ−1 leads to aLagrangian of the form (4.19) with λ = 0, but with the analogue of Xz entering quadratically. Unfortunately, in thereal case this trick does not work because the characteristic polynomial is of order six instead of three, meaning that〈ψ〉 can be expressed in terms of 〈ψ−n〉, where n goes up to five. This alternative constraint can therefore only makethe equation of motion of ψ worst, i.e. of higher order than the quadratic one (4.14).

– 8 –

5 Derivation of the constraint-free BF action

Here we find it more convenient to consider as our starting point the following Plebanski Lagrangian

L = 〈B|F 〉 − 1

2〈B|ψ|B〉+ φ

[⟨z2ψ

⟩− λ], (5.1)

which is obtained from (4.3) with the redefinitions

|B〉 → z−1|B〉 , ψ → zψz . (5.2)

Following [18], we consider a redefinition of the form

|B〉 → χ1|B〉+ χ2|F 〉 , (5.3)

where χ1,2 are such that the resulting Lagrangian maintains its “canonical” BF form, i.e. the ∼BF term is simply 〈B|F 〉 and the ∼ FF term arises only through the topological combinations〈F |w−2|F 〉/2, for some constant invariant matrix w, which we neglect anyway. We find it actuallymore convenient to proceed in two steps. First, we demand that the resulting matrices in 〈B| . . . |F 〉and 〈F | . . . |F 〉/2 be the same invariant matrix w−2, so that the corresponding conditions read

χ1 −1

2[χ1ψχ2 + χ2ψχ1] = w

−2 , χ2 −1

2χ2ψχ2 =

1

2w−2 , (5.4)

and admit the symmetric solutions

χ1 = w−1 ±1√1−ψw

w−1 , χ2 = w−1 1∓√1−ψw

ψw

w−1 , (5.5)

whereψw := w−1ψw−1 . (5.6)

The resulting Lagrangian reads

L→ 〈B|w−2|F 〉 − 1

2〈B|w−1 ψw

1−ψw

w−1|B〉+ φ[〈z2ψ〉 − λ

], (5.7)

so we perform one more redefinition

|B〉 → w2|B〉 , ψ → wψw

1+ψw

w , (5.8)

to obtain the canonical BF form

L→ 〈B|F 〉 − 1

2〈B|ψ|B〉+ φ

[⟨(zw)

2 ψw

1+ψw

⟩− λ

]. (5.9)

The Plebanski Lagrangian now corresponds to the w−1 → 0 limit. For finite w, however, (5.9) is nolonger linear in ψ, so the latter can be integrated out without constraining other fields. Its equationof motion can be put in the form (4.16)

(1+ψw)wYw (1+ψw) = 2φ (zw)2, Y := |B〉〈B| , (5.10)

and is therefore solved using (4.17)

ψ = w[±√2φzw

(zw2Yw2z

)−1/2wz − 1

]w , (5.11)

leading to

L→ 〈B|F 〉 + 1

2〈wYw〉 ∓

√2φ⟨√

zw2Yw2z⟩+ φ

[⟨(zw)

2⟩− λ]. (5.12)

– 9 –

Finally, integrating out φ, whose solution reads

√2φ = ±

⟨√zw2Yw2z

⟩⟨(zw)

2⟩− λ

, (5.13)

we get the constraint-free BF Lagrangian

L→ 〈B|F 〉 + 1

2

〈wYw〉+

⟨√zw2Yw2z

⟩2

λ−⟨(zw)

2⟩

, (5.14)

hence the action (1.6). As in the pure connection case, we can use (4.18) to write this as

L ≡ 1

2

Bab ∧ F ab +

⟨w2Y

⟩+

⟨√z2w4Y

⟩2

λ−⟨(zw)

2⟩

, (5.15)

at the expense of having non-symmetric matrix arguments. Note that, contrary to the pure connectioncase, here we do have access to the λ = 0 theory. We still have a matrix square root leading to inversematrices in the equations of motion, but now this concerns Y , which is invertible on the Minkowskisolution because Bab is essentially the vierbein.

As a last check of the pure connection Lagrangian (4.22), we can now also integrate out |B〉. Itsequation of motion reads

w2 +

⟨√zw2Yw2z

⟩

λ−⟨(zw)

2⟩ zw2

(zw2Yw2z

)−1/2w2z

|B〉 = −|F 〉 , (5.16)

so we must express Y in the square bracket in terms of X ≡ |F 〉〈F | in order to invert this. We firsttake the outer product of each side with itself, multiply by z from right and left to form a total squareand then take the square root

√X z = ±

√zw2Yw2z + (zw)

2

⟨√zw2Yw2z

⟩

λ−⟨(zw)

2⟩

. (5.17)

Next, taking the trace of this⟨√

X z

⟩= ± λ

⟨√zw2Yw2z

⟩

λ−⟨(zw)

2⟩ . (5.18)

and plugging the result back inside (5.17), we obtain

√zw2Yw2z = ±

[√X z − 1

λ

⟨√X z

⟩(zw)

2

], (5.19)

which, once plugged inside (5.16), allows us to isolate |B〉

|B〉 = w−1

⟨√X z

⟩1− λ (zw)

−1 √X z (zw)

−1

λ (zw)−1 √X z (zw)−1 w−1|F 〉 . (5.20)

Finally, plugging this back inside (5.14), and using also the shortcut (5.19), we recover the pureconnection Lagrangian (4.22), up to topological terms ∼ F 2. The latter are −〈w−1Xw−1〉/2, thusprecisely canceling the 〈F |w−2|F 〉/2 term that was introduced by the redefinition (5.3). The resultingpure connection Lagrangian is therefore consistently independent of w, since the latter controls theredefinition of a field that has been integrated out.

– 10 –

6 The SL(2,C) and SD formulations

Let us now express the Lagrangians (4.22) and (5.14) in terms of the universal covering group SL(2,C),in which case the independent variables are the self-dual components

Ai := −1

2εijkAjk + iA0i , (6.1)

F i := −1

2εijkF jk + iF 0i ≡ dAi +

1

2εijkAj ∧Ak , (6.2)

Bi :=1

2

[−1

2εijkBjk + iB0i

], (6.3)

and their complex conjugates, and we use i, j, k, . . . to denote the spatial Lorentz indices that aredisplaced with δij . In terms of these quantities the Lagrangians are formally the same as (4.22) and(5.14), only now

⋆ :=

(iδij 00 −iδij

), X :=

1

2

(F i ∧ F j F i ∧ F j

F i ∧ F j F i ∧ F j

), Y := 2

(Bi ∧Bj Bi ∧ Bj

Bi ∧Bj Bi ∧ Bj

), (6.4)

which are 6× 6 complex symmetric matrices. In particular, a real invariant matrix z becomes

z ≡(zδij 00 zδij

), z := z|1→1,⋆→i . (6.5)

To prove (6.4) one must simply check that all the traces {〈Xnz〉}n=1,...,6 coincide, because these

numbers fully determine the characteristic polynomial of X z, and thus its eigenvalues, which in turnfully determine L. The same then holds for the functions of Y . With the SL(2,C) forms at hand,we can now recover the SD Lagrangians in their pure-connection and BF forms. We start with theformer, which is given by [10, 11]

LSD[F ] :=i

2λ

⟨√X⟩2

, X ij := F i ∧ F j , (6.6)

and is supplemented by the reality constraints [38]

F i ∧ F j = 0 , Re

[⟨√X⟩2]

= 0 , (6.7)

with the latter making the Lagrangian real. Note that the complex metric defined by [38, 39]

gαβ ∼ εijk εµνρσF i

αµFjνρF

kσβ ,

√|g| d4x =

λ

Λ2LSD , (6.8)

is the one satisfying the Einstein equations when Ai is a solution [38], and the ten reality constraints(6.7) are equivalent to the statement that this metric is real Lorentzian. Since this property ispreserved under evolution, these constraints are compatible with the dynamics from the canonicalviewpoint.

The easiest way to obtain (6.6) from the real Lagrangian is to simply set γ = −i in (4.23) with(6.4), thus projecting out the anti-self-dual component

z2 →(2iδij 00 0

)⇒ L→ i

2λ

⟨√X⟩2

. (6.9)

Another option is to impose the reality constraints (6.7) on (4.23) using (6.5) for generic γ ∈ R. Thefirst constraint of (6.7) makes X block-diagonal, so

√z2X → 1√

2

(z√X 0

0 z√X

)⇒ L→ 1

4λ

⟨z√X + z

√X⟩2

. (6.10)

– 11 –

Then, rewriting the second constraint of (6.7) as

Re⟨√X⟩= ±Im

⟨√X⟩

⇒⟨√X⟩± i⟨√X⟩= 0 , (6.11)

we find, depending on the choice of sign,

L+ → i

2λ

⟨√X⟩2

, L− → −i2γ2λ

⟨√X⟩2

. (6.12)

Thus, with the plus sign we recover (6.6) indeed, while with the minus sign the constants are rescaledas in (4.9), meaning that this corresponds to the wrong sector of the theory. It is therefore moreaccurate to state (6.11) with “+” as the scalar constraint leading to GR, rather than the one in (6.7).

We next want to derive the BF form without simplicity constraints in the SD case. Note that,contrary to the SD Plebanski Lagrangian, where the reality constraints are expressed exclusivelyin terms of Bi, here this is no longer the case, because that field has been mixed with F i in theredefinition

Bi → χij1 B

j + χij2 F

j . (6.13)

For this reason, here we will only consider the easier approach which is to simply set γ = −i. Asin the pure-connection Lagrangian, the resulting z2 projects out all Bi dependencies in the matrixsquare root term of (5.15), while in the other two terms the anti-self-dual components are alreadydecoupled. We can therefore collect the SD sector to find

LSD[B,F ] := Bi ∧ F i + w2

[〈Y 〉+ 2iw2

λ− 6iw2

⟨√Y⟩2]

, Y ij := Bi ∧Bj , (6.14)

where w is the complex number in w according to the relation (6.5). The above Lagrangian is indeedthe Lorentzian version of the one found in [17, 18] which is obtained by sending λ→ iλ.

7 The linearized pure connection action over (A)dS

Let us now consider the (A)dS solution Aab, in which case there exists some vierbein ea such that

Dea ≡ dea + Aab ∧ eb = 0 , F ab =

αΛ

3ea ∧ eb , (7.1)

where α = 1 for the solution in the right sector and α = −γ in the wrong one, given the relation (4.9)between the two. In particular, the matrix entering the Lagrangian (4.22) is

X z = −α2Λ2e

9z2⋆ ≡ α2Λ2e

9

(γ2 − 1

2γ21− 1

γ⋆

). (7.2)

We next consider a perturbation around that solution

δAab := Aab − Aab , δF ab := F ab − F ab ≡ D δAab + δAac ∧ δAcb , (7.3)

and compute the part of the Lagrangian that is second order in δAab, i.e. the linearized theory. Inorder to expand the Lagrangian in powers of δAab, we need to expand the trace of a matrix squareroot around X z. However, the latter is not a multiple of the identity when γ is finite, so we cannotsimply Taylor expand without caring about the order of the matrices. We therefore require a bit morealgebra before we can begin. We first decompose X z into irreducible pieces (2.21)

X z ≡ 1

6

(S1− S ⋆

)+ S+ + S− , (7.4)

and note that the last two are first order in δAab, since the background (7.2) is an invariant matrix,

so we can expand the Lagrangian to second order in S+ and S−. The desired expression is found bytaking the most general symmetric ansatz containing the involved matrices

√X z = a1+ a ⋆+ a+S+ + a+ ⋆ S+ + a−S− + a++ S

2+ + a++ ⋆ S

2+ +

– 12 –

+ a−−S2− + a−− ⋆ S2

− + a+−

(S+S− + S−S+

)+O(S3

±) , (7.5)

squaring both sides and solving for the coefficients, which therefore depend on S and S. Taking thetrace of the result and then the square, only four combinations survive

⟨√X z

⟩2≡ c− 9

2

⟨c+S

2+ + c−S

2−

⟩+O(S3) . (7.6)

Indeed, all first order terms drop thanks to the tracelessness of S±, S±S∓ and ⋆S±S∓ are odd, sotheir trace vanishes too, and we have used the cyclicity of the trace to also get

⟨⋆S2

−

⟩≡ −〈S− ⋆ S−〉 ≡ −

⟨⋆S2

−

⟩⇒

⟨⋆S2

−

⟩≡ 0 . (7.7)

The coefficients c, c+ and c− in (7.6) are determined up to a sign ambiguity, e.g.

c = 3(S ±

√S2 + S2

), (7.8)

which, in fact, distinguishes the right and wrong sectors. To determine which is which, we firstevaluate the Lagrangian on the (A)dS solution (7.2) to find

L[A] =c

16πGΛ≡ α2Λe

16πG

γ2 − 1±(γ2 + 1

)

γ2, (7.9)

and compare with the original Lagrangian (4.1) evaluated on the same solution (7.1)

L[e, A] =Λe

8πG. (7.10)

We see that the right sector α = 1 corresponds to the plus sign in (7.9), whereas the wrong sectorα = −γ, which is related to (7.10) by the replacement (4.9), is consistently reproduced by taking theminus sign in (7.9). Focusing on the right sector from now on, the coefficients in (7.6) are given by

c = 3(S +

√S2 + S2

), c+ =

(S 1+ S ⋆

)2+(S 1+ S ⋆

)√S2 + S2

(S2 + S2

)3/2 , c− =2√

S2 + S2.

(7.11)As far as the O(δA2) part of the Lagrangian is concerned, c+ and c− can be directly evaluated onthe background solution (7.2)

c+ ≡ 6γ3

Λ2e (γ2 + 1)3

[γ(γ2 − 3

)1+

(3γ2 − 1

)⋆], c− ≡ 6γ2

Λ2e (γ2 + 1), (7.12)

since they multiply quantities that are already O(δA2). As for c, the most general form of thesecond-order part in δAab is

c(2) ≡ α1 〈⋆X (1)〉2 + 2α2 〈⋆X (1)〉〈X (1)〉+ α3 〈X (1)〉2 + β1 〈⋆X (2)〉+ β2 〈X (2)〉 , (7.13)

but the ∼ β1,2 terms do not contribute to the equations of motion, because they correspond to thesecond-order part of the topological terms 〈F |(β1 ⋆ + β2 1)|F 〉 which we neglect here. It then turnsout that for this particular c(S, S) function (7.11) the α1,2 terms vanish as well, leaving us with

c(2) → 9

8Λ2e

γ2 + 1

γ2〈X (1)〉2 . (7.14)

We thus need to compute 〈X 〉, S+ and S− to linear order in δAab. In particular, we want to expressthese in terms of the components of the curvature 2-form F ab in the background space-time basis ea

Fabcd := eµa eνbFµνcd , Fab := Fc

acb , F := Faa , (7.15)

– 13 –

so that we can now treat Fabcd as a non-symmetric matrix and in particular

F ≡ Λ

31 , X ≡ −Fκ ⋆F e . (7.16)

Perturbing this expression, and decomposing δF into irreducible components (2.21)

δF ≡ 1

6

(S1− S ⋆

)+ S+ + S− +A+ +A− , (7.17)

we find

δX ≡ −Λe

3[⋆ δF + δFκ⋆] ≡ −2Λe

3

[1

6

(S1+ S ⋆

)+ ⋆

(S+ +A−

)], (7.18)

and thus

δX z ≡ −2Λe

3

[1

6z2(S1+ S ⋆

)+ ⋆

(z2S+ + |z|2A−

)], |z|2 ≡ γ2 + 1

2γ2. (7.19)

The quantities of interest are therefore

〈X 〉 ≡ −2Λe

3S +O(δA2) , S+ ≡ −2Λe

3z2 ⋆ S+ +O(δA2) , S− ≡ −2Λe

3|z|2 ⋆A− +O(δA2) ,

(7.20)so plugging these inside (7.6) and using (7.12) and (7.14) gives

(⟨√X z

⟩2)(2)

≡ c(2) + 2Λ2e2⟨c+z

4S2

+ − c−|z|4A2−

⟩(7.21)

≡[γ2 + 1

2γ2S2 − 3

⟨(1− ⋆

γ

)S

2

+ +γ2 + 1

γ2A2

−

⟩]e ,

and we must finally compute S, S+ and A− in terms of δF through (7.17). We find

S ≡ 〈⋆C〉 , S+ ≡ 1

2

[C + Cκ +

1

3〈⋆C〉 ⋆

], A− ≡ 1

2[C − Cκ] , (7.22)

where

Cabcd := Fabcd −1

2(ηacFbd − ηbcFad − ηadFbc + ηbdFac) +

1

6κabcdF , (7.23)

is traceless, having therefore zero background, but without the symmetries of the Weyl tensor. Thefinal result can then be put in the following form

S(A)dS = − 3

64πGΛ

∫e

[γ2 + 1

γ2Cabcd Ccdab − 1

γ2S+abcd Sabcd

+ − 1

γCabcd Cabcd

]+O(δA3) , (7.24)

where

S+abcd ≡ 1

3

[Cabcd +

1

2(Cacbd − Cadbc − Cbcad + Cbdac) + Ccdab

], Cabcd :=

1

2ε efab Cefcd , (7.25)

satisfy the symmetries of the Weyl and Riemann tensor, respectively.2 As one could expect, a finiteImmirzi parameter leads to a parity violating term ∼ Cabcd Cabcd, while in the parity-symmetric case|γ| → ∞ we recover the result of [19, 20]

Snon−chiral(A)dS = − 3

64πGΛ

∫e Cabcd Ccdab +O(δA3) . (7.26)

2Indeed, Cabcd ≡ Ccdab is shown by contracting both sides with εcdef and Ca[bcd] ≡ 0 is shown by contracting Cabcd

with εbcde. Note, however, that Cabcd is not traceless, although it is effectively made so in the action by being contractedwith a traceless tensor.

– 14 –

On the other hand, given the symmetries of Cabcd and the tracelessness of Cabcd, in the (anti-)self-dual

cases γ = ±i the action depends exclusively on the irreducible component S+

S(A)SD(A)dS = − 3

64πGΛ

∫eW±

abcdWabcd± +O(δA3) , (7.27)

and only through the combination

W±abcd :=

1√2

[S+abcd ±

i

2ε efab S+

efcd

]. (7.28)

Finally, the fact that the action depends solely on the traceless component Cabcd, and not on Fab, is apeculiarity of GR. To see this, let us consider the generalization of (4.22) to an arbitrary Lagrangianfunction

L = f(X , ⋆) , (7.29)

that is a homogeneous function of X of degree one

f(cX , ⋆) ≡ cf(X , ⋆) . (7.30)

This requirement is essentially the condition of L being a 4-form, or equivalently of having a La-grangian density of weight one, i.e. it is needed for invariance under the full diffeomorphism group.Repeating the above procedure for (7.29), we first note that we can again focus on squares of X (1),

because the ∼ X (2) contributions correspond to topological terms. Then (7.18) implies that the result

can only depend on the S, S, S+ and A− components of the perturbation δF . As we just saw, all ofthem depend exclusively on the fully traceless component Cabcd, except for S ≡ 〈δF〉 ≡ δF/2. Thus,it is the fact that the c function is independent of S that makes the quadratic GR action special.

8 No go for real modified actions

A remarkable fact about the SD formulation is that it admits an infinite parameter family of modi-fications with the same degrees of freedom as GR [21–29]. In its pure connection formulation this issimply the SD analogue of (7.29), that is, all the Lagrangians of the form

L = f(X) , X ij := F i ∧ F j , (8.1)

where f is homogeneous of degree one, thus generalizing (6.6). As for the reality constraints, a naturalgeneralization of (6.7) would be [38]

F i ∧ F j = 0 , Im f(X) = 0 , (8.2)

where the latter makes again the action real. Unlike in the case of GR, however, these reality con-straints (8.2) are not guaranteed to be compatible with the dynamics of (8.1) [38] and, in fact, it hasbeen claimed that they are not [40]. If this is the case, then the analogous metric to (6.8)

gαβ ∼ εijk εµνρσF i

αµFjνρF

kσβ ,

√|g| d4x =

1

m4f(X) , (8.3)

where m is some mass scale, cannot be real Lorentzian throughout the evolution. There might existdifferent reality constraints than (8.2) that are compatible, and a different metric than (8.3) thatthese constraints would make real Lorentzian, but if not, then these modified theories admit no realLorentzian metric formulation and therefore would not count as theories of gravity. Nevertheless,prior to imposing the reality constraints, the Lagrangian (8.1) has the same degrees of freedom as(6.6). Consequently, in the Euclidean case where Ai is real, (8.3) is real Euclidean and no constraintsare required, these theories are genuine modified (Euclidean) gravity theories.

Putting aside this yet undetermined aspect for the Lorentzian case, the existence of this familyof modified massless spin-2 self-interactions is possible because the standard uniqueness theorem of

– 15 –

GR assumes symmetry under parity transformations, which is not the case of theories based on theself-dual component alone [41]. In contrast, therefore, the analogous modification in the real non-chiral case L = f(⋆X ) has eight degrees of freedom in general [42], namely, one massless graviton(2), one massive graviton (5) and a scalar (1) [43, 44], meaning this is actually a “bigravity” theory.This degree of freedom count remains unchanged if we allow for real parity-violating terms, i.e. thegeneral Lagrangian (7.29), because in the absence of imaginary factors both the self-dual and theanti-self-dual components will be present and thus generically active. Note that this is not true in theEuclidean case, because the corresponding irreducible representations of SO(4) are real, so one canproject out one of the two gravitons using only real parameters [44]. Note also that, at the level ofthe BF formulation, the corresponding generalization of (5.14) takes the form

L = 〈B|F 〉 − 1

2V (Y , ⋆) , (8.4)

for some “potential” V that is also homogeneous of degree one. Integrating out |B〉 then leads to aLagrangian of the form (7.29).

Here we would like to point out that, in the real Lorentzian formulation of these modified theories,one of the two gravitons is necessarily a “ghost”, i.e. it has negative-definite kinetic energy. Thecorresponding QFT is therefore either unstable or non-unitary and, more generally, such a pathologycasts doubt on whether the fully non-linear theory can lead to sensible dynamics, both at the classicaland quantum levels. Note that in [44] the focus was on the scalar, which is notoriously ghost-like inthe case of standard bigravity [45], and on ways to eliminate it from the spectrum, in analogy withstandard bigravity where this is possible [46–50]. However, if one of the two gravitons is a ghosttoo, then the only viable theory is the one with no scalar and no second graviton, that is, GR. Ourargumentation will also show why the second graviton was not identified as a ghost in [43, 44].

In order to prove our assertion, we first need to lay down some quick facts. Let us considerthe linearized theory for the fluctuations δAab around some background solution Aab with associatedtorsionless vierbein ea. Focusing on spin-2 excitations (gravitons), we note that these lie in symmetric3-tensors and that one can form exactly two such fields out of δAab

hij := eµ(iδAj)0µ , hij :=

1

2eµ(iεj)klδAkl

µ . (8.5)

The equations of motion of the pure connection theory being second-order in derivatives, there are atmost two gravitons. In the case of GR, the simplicity constraints imposed by ψ on |B〉, and thereforeultimately on the dynamics of the connection, neutralize a combination of hij and hij , thus leavingus with a single graviton. From the viewpoint of the canonical formulation, this corresponds to thepresence of second-class constraints [35, 42] which are therefore due to the particular structure of theaction, not to its gauge symmetries. The necessity of such constraints can be understood from thefact that the gauge parameters of the theory do not contain symmetric 3-tensors, so the correspondingfirst-class constraints and possible gauge choices cannot completely eliminate components of the form(8.5). As already noted in the previous section, at the level of the linearized action (7.24), the specialstructure of GR manifests itself in the fact that the dynamics only depend on part of the curvaturecomponents Fabcd, namely, the fully traceless tensor Cabcd.

Thus, in the non-GR cases where the Lagrangian does not have the specific structure (4.22) or(5.14), one should generically expect both hij and hij to contain degrees of freedom, as shown at thefully non-perturbative level in [42]. In contrast, these extra degrees of freedom cannot occur in the SDcase, simply because one can only form a single symmetric 3-tensor out of the self-dual component of

the connection alone eµ(iδAj)µ . From the canonical viewpoint, SO(3,C) GR only contains first-class

constraints that are related to the gauge symmetries. Since the latter are the same for all Lagrangians(8.1) and the number of reality constraints (8.2) is also the same, the degree of freedom count is thesame as in GR.

Now note that hij and hij are even and odd under parity, respectively. Thus, if the action isparity-symmetric, then its free part will be diagonal in hij and hij , while if it is parity-violating, thetwo gravitons will lie in two independent linear combinations of these two fields. In any case, however,

– 16 –

the fact that hij and hij correspond to a boost and a rotation component of δAab, respectively, impliesthat the kinetic terms of the two independent combinations, along with their associated momenta,will enter with opposite signs in the free Hamiltonian. This is because the Killing form of the Lorentzalgebra has split signature diag(−1,−1,−1, 1, 1, 1) so, whatever the independent combinations of hij

and hij are, they cannot enter through a positive definite scalar product. As a result, even aftertaking into account all available constraints and all possible gauge fixings, the free Hamiltonian of themodified theories cannot be positive-definite. In particular, if one graviton has positive-definite kineticenergy, then the other one will have negative-definite kinetic energy. This does not occur in standardbigravity, because in that case one has full control over the kinetic terms of the two metric/vierbeinfields. In contrast, here both spin-2 excitations are forced to enter through a single spin connectionfield, which incidentally forces them to have the opposite behavior under parity transformations,contrary to the two gravitons of standard bigravity. The reason why this important point was missedin the literature is probably the fact that the explicit studies of the linearized theories performed sofar [43, 44] were always carried out in Euclidean signature, where the Killing form has definite sign∼ diag(1, 1, 1, 1, 1, 1), and that the Lorentzian case is usually approached through complex GR.

The fact that the modifications (7.29) are pathological in the real case is important for tworeasons. First, it highlights in yet another way the special place GR holds in describing interactingspin-2 dynamics. Second, such modified theories have been considered as a starting point for unifiedgauge theories where the gauge group is extended to include other forces on top of gravity [51, 52].Since the purely gravitational theory is non-viable to begin with, such generalizations are bound to failtoo, i.e. they lack the action structure that is required to eliminate the ghost graviton. Consequently,if one wishes to extend the group using a modified gravity theory, then only the SD theories (8.1)would seem appropriate, as considered in [53, 54] for instance. In that case, however, there remainsthe difficulty of finding reality constraints that are compatible and that allow the construction of somereal Lorentzian metric.

9 Group extensions and unification

The fact that only a particular choice of Lagrangian function leads to sensible dynamics at the purelygravitational level constraints severely the set of potentially viable actions for the correspondingunified theories, i.e. those obtained through extension of the gauge group SO(1, 3) → G. Indeed, ifthe G = SO(1, 3) case has ghosts, then so will the G ⊃ SO(1, 3) one. Conversely, extending the groupwith a Lagrangian that has the GR form has a chance of yielding a viable theory. So let us discusssome aspects of these theories.

We first observe that the presence of the ⋆ operator precludes us from extending the gauge groupin general. The only exceptions are groups that also have such a complex structure, e.g. groupsthat are the complexification of some other group. This is also clear from the SL(2,C) form of theLagrangian, i.e. (4.22) with (6.4), which is straightforwardly generalizable for groups with complexcurvature coefficients. Note, however, that for the specific chiral choices γ = ±1 we have z2 = ±1, sothe pure connection Lagrangian (4.23) is independent of ⋆, and so is the BF one (5.15) if we restrictto w = w1. Observe also that the difference between the two γ = ±1 choices amounts to flipping thesign of λ, so from now on we consider the case γ = 1 for definiteness. Moreover, the right and wrongsectors too are now related by a simple sign flip of λ (see (4.9)).

With |γ| = 1 the GR actions can be generalized straightforwardly to the ones of arbitrary groupG by simply keeping the same matrix/vector notation, only now the implicit indices are A,B,C, . . .of the adjoint representation of G, with structure constants fA

BC , and are displaced using the Killingform

κAB ∼ fCDAf

DCB . (9.1)

Interestingly, with the choice G = SO(3), or equivalently SU(2), we obtain the pure-connectionLagrangian for Euclidean four-dimensional GR derived by Krasnov [10, 11] as a stepping stone forthe SD Lorentzian one. We now consider the relevant case for unification G ⊃ SO(1, 3), meaningin particular that |G| := dim(G) > 6, and work with the φ,ψ-dependent BF formulation (5.9), now

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reading

L = 〈B|F 〉 − 1

2〈B|ψ|B〉+ φ

[⟨w2ψ

w21+ψ

⟩− λ

]. (9.2)

The equations of motion of |B〉 and ψ read

|F 〉 = ψ|B〉 ,(w21+ψ

)Y(w21+ψ

)= 2w4φ1 , (9.3)

respectively, and with the former we can rewrite the latter as

|K〉〈K| = 2w4φ1 , |K〉 := w2|B〉+ |F 〉 . (9.4)

Invoking some arbitrary non-singular vierbein field ea, defining

KAab :=

1

2(√⋆) cd

ab eµc eνdK

Aµν , (9.5)

and using a matrix/vector notation only for the [ab] indices of that quantity, we can express (9.4) as

〈KA|KB〉 e = −2w4φκAB . (9.6)

Now if we assume that G is semi-simple, then there exists a generator basis such that κAB is diagonalwith ±1 entries. In that case (9.6) admits no solutions, because it states that there exist |G| > 6six-dimensional vectors |KA〉 6= 0 that are linearly independent with respect to the scalar productκ[ab][cd]. The gauge group of the Standard Model is semi-simple, so if it is to be unified with SO(1,3)within some larger group, one would expect the Jordan form of the corresponding κAB to contain atleast one diagonal block with dimension larger than six, thus making again (9.6) impossible to solve.At the level of the pure-connection (4.22) and constraint-free BF (5.14) formulations, this problemmanifests itself as the impossibility of X and Y to be invertible matrices for |G| > 6. They both takethe respective forms 〈FA|FB〉 and 〈BA|BB〉, but the vectors forming the columns of these matricescannot be linearly independent for |G| > 6. Note that the absence of classical solutions does notnecessarily mean that the quantum theory does not exist, it only means that the path integral hasno saddle point. Without a notion of “on-shell”, however, most of the perturbative mathematicalmethods fail, thus making the manipulation of such theories considerably more involved.

This situation can be avoided by introducing an extra set of dynamical 0-forms hα, forming somerepresentation R of G, along with their corresponding covariant momenta, i.e. a set of 3-forms πα,given that we are working with first-order Lagrangians. Using the generators in this representation(TA

R )αβ, one could then form a symmetric matrix of 0-forms z ≡ z(h) and consider for exampleLagrangians that follow the GR structure (5.9)

L = 〈B|F 〉+ πα ∧Dhα − 1

2〈B|ψ|B〉+ φ

[⟨z2(h)

w2ψ

w21+ψ

⟩− λ(h)

]. (9.7)

The analogous equation to (9.4) would then be

|K〉〈K| = 2w4φz2(h) , (9.8)

and would therefore admit solutions where z2 has rank six at most. In particular, the vacuum solutionof physical interest, where space-time exists and the extra forces are trivial, would correspond to theVEV of z2 being some pure-Lorentz invariant matrix, thus providing the desired dynamical symmetrybreaking mechanism G → SO(1, 3) × G′. Observe also that it is necessary to make hα dynamical,because otherwise it would alter the ψ-dependence of the action, thus introducing ghosts. In fact,although we have maintained the GR structure in this extension, it is not at all guaranteed that aLagrangian of the form (9.7) maintains the desired second-class constraints, or that it is devoid ofpathologies in general, and a dedicated study should be carried out. We leave this to future work.

Finally, note that the presence of fields like hα and πα is also welcome for another reason, namely,because they allow us to include the type of fields one encounters in the matter sector of the StandardModel. Indeed, on the one hand the Higgs field is a set of 0-forms, while 3-forms with non-zero VEVare needed in order to form first-order Lagrangians out of spinor 0-forms ∼ π ∧ ψDψ.

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Acknowledgments

The author is grateful to the referees for useful comments and suggestions. This work is supportedby a Consolidator Grant of the European Research Council (ERC-2015-CoG grant 680886).

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